Geology Reference
In-Depth Information
Fig. 5.7
Conventions and
geometry of a
two-dimensional problem.
The polygon
approximating the body
cross-section has vertices
v
i
, which are ordered
clockwise
0
4 G¡
M
g
D
0
¦
4 G¡
H
g
Therefore, for any scalar field ¥ that is con-
stant in the
y
direction, the directional derivative
of
¥
along direction
n
results to be:
V
D
(5.40)
where ¦ is the magnetic susceptivity and
H
is
the inducing geomagnetic or paleomagnetic field.
This expression is called
Poisson's relation
.It
states that the magnetic potential of a uniformly
magnetized body having constant density is pro-
portional to the component of the gravity field in
the direction of magnetization. Therefore, taking
the gradient of Eq. (
5.40
) we obtain that the
anomalous field of a body in a 2-dimensional
problem can be written as:
@¥
@n
Dr
¥
n
D
@¥
@x
cos I sin '
C
@¥
@
z
sin I
(5.43)
It is easy to determine the vertical and horizon-
tal components of the anomalous field combining
expressions (
5.41
)and(
5.43
). We have:
0
M
4 G¡
@g
x
@n
F
x
D
0
M
4 G¡
@
@n
D
0
¦H
4 G¡
@
@n
(5.41)
@g
x
@x
cos I sin '
C
sin I
F
Dr
V
D
0
M
4 G¡
@g
x
@
z
D
(5.44)
where
n
is the direction of induced or remnant
magnetization. This solution implies that unlike
the gravity anomaly, the magnetic anomaly also
depends on the strike of the body, as this affects
the direction of magnetization. Let us assume that
the body magnetization is purely NRM. Let
I
and
' be respectively the mean paleomagnetic field
inclination in the survey area, and the strike of the
body measured counterclockwise from the paleo-
magnetic North to the negative
y
-axis (Fig.
5.7
).
The unit vector associated with the magnetiza-
tion direction can be written as:
8
<
0
M
4 G¡
@g
z
@n
F
z
D
@g
z
@x
cos I sin '
C
sin I
0
M
4 G¡
@g
z
@
z
D
(5.45)
Quantities
g
z
and
g
x
can be calculated respec-
tively using (
5.34
) (because
g
z
”) and a similar
formula for
g
x
. These formulae can be written as
follows:
N
N
n
x
D
cos I sin '
n
y
D
cos I cos'
n
z
D
sin I
X
X
g
x
D
2G¡
X
n
I
g
z
D
2G¡
Z
n
(5.46)
(5.42)
:
nD1
nD1
